10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS 389
It is immediately verified that if U is a hermitian (resp. positive hermitian,
resp. unitary) operator on a Hilbert space E, and if S is an isomorphism of E
onto a Hilbert space E', then the operator SUS'1 on E' has the same
property (since these properties involve only the Hilbert structure). Hence
we may take T(u) = M^(u), and in this case the sufficiency of the conditions
stated is clear. On the other hand, if for example there exists a measurable
subset X of K such that ju(X) = a > 0 and J(u(x)) ^ j8 > 0 for all x e X, then
we have </((T(w) - <pxl<Px)) = s(\ udp\^afl and therefore T(u) is not
hermitian.,The other cases are dealt with similarly.
(15.10.6) (i) The orthogonal projectors belonging to the algebra
are the operators of the form T((px), where X is a universally measurable subset
ofK.
(ii) T(^C(K)) is a maximal commutative subalgebra of&(E).
(iii) A closed vector subspace EofE is stable under T if and only if it is of
the form T(<px)(E), where X is a universally measurable subset o/K.
(i) By virtue of the characterization of orthogonal projectors on E
(15.5.3.1) and by (15.10.5), it follows that T(u) is an orthogonal projector if and
only if u is almost everywhere equal to a /(-measurable bounded real-valued
function v such that v2 = v. Hence v = <px, where X is a /(-measurable subset
of K, and the result follows.
(ii) It is clearly enough to prove that a continuous operator Fe JS?(E)
which commutes with T(u) for all ue ^C(K) is of the form T(v). If a is a
totalizing vector for T, then for each universally measurable subset X of K we
have
*)V-a\a)\ = \(VT(q>d-a\T(q>d-a)\
iin<px)^ii2 = iim- [<pxdnata = \\v\\-
because T(<px) is an orthogonal projector which commutes with V. The
Lebesgue-Nikodymtheorem (13.15.5(cy)) therefore shows that uv.a}0 = h • n
for some /x-integrable function h. Since also it follows from the inequality
above that
for every step function we^c(K), we conclude that the linear form
w h-* I hw d\i defined on the vector space of these step functions extends bynd w are in ^C(K), tnen f°r